The embedding \(W^{n,1}\) into \(C_0\) on compact Riemannian manifolds
The authors are concerned with the Sobolev embedding of \(W^{n,1}(\mathbb{R}^n)\) (= the space of functions whose derivatives of order lower than \(n\in \mathbb{N}^{*}\) belong to \(L^1\)) into \(C_0(\mathbb{R}^n)\) (= the space of bounded continuous functions). The corresponding inequality asserts that there exists \(C>0\) such that for all \(u\in C_c^{\infty}(\mathbb{R}^n)\), \[ \| u\| _{\infty}\leq C\int_{\mathbb{R}^n} | \nabla^n u| \,d\xi, \tag{S(C)} \] where \(\xi\) stands for the standard Euclidean metric on \(\mathbb{R}^n\) and where \[ | \nabla^n u| :=\sqrt{\sum_{i_1, \dots,i_n\in \{1,\dots,n\}} \left(\frac{\partial^n u}{\partial x_{i_1}\cdots\partial x_{i_n}} \right)^2 }. \] They compute the values of the best constants in S(C) for \(n\in \{1,2\}\). The previous inequality is adapted for smooth compact Riemannian manifolds of dimension \(n\geq 1\), which involves two constants, and its optimality is discussed. Similar results concerning inequalities involving derivatives of order one have been obtained by \textit{T. Aubin} and \textit{Y. Y. Li} [J. Math. Pures Appl., IX. Sér. 78, 353--387 (1999; Zbl 0944.46027)], \textit{O. Druet} [Math. Ann. 314, 327--346 (1999; Zbl 0934.53028)], \textit{E. Hebey} and \textit{M. Vaugon} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13, 57--93 (1996; Zbl 0849.53035)] and the first author [Proc. R. Soc. Edinb., Sect. A, Math. 131, 621--646 (2001; Zbl 1040.53048)].
scientific article; zbMATH DE number 1968877